Abstract : Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. Let $\mathcal{L}=-\Delta_{\mathbb{H}^{n}}+V$ be the Schr\"{o}dinger operator on $\mathbb{H}^{n}$, where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q_{1}}$ for $q_{1}\geq Q/2$. Let ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ be the Hardy space associated with the Schr\"{o}dinger operator $\mathcal{L}$ for $Q/(Q+\delta_{0})
Abstract : Although machine learning shows state-of-the-art perfor\-man\-ce in a variety of fields, it is short a theoretical understanding of how machine learning works. Recently, theoretical approaches are actively being studied, and there are results for one of them, margin and its distribution. In this paper, especially we focused on the role of margin in the perturbations of inputs and parameters. We show a generalization bound for two cases, a linear model for binary classification and neural networks for multi-classification, when the inputs have normal distributed random noises. The additional generalization term caused by random noises is related to margin and exponentially inversely proportional to the noise level for binary classification. And in neural networks, the additional generalization term depends on (input dimension) $\times$ (norms of input and weights). For these results, we used the PAC-Bayesian framework. This paper is considering random noises and margin together, and it will be helpful to a better understanding of model sensitivity and the construction of robust generalization.
Abstract : In this paper, we are concerned with a class of mixed Hessian curvature equations with non-degeneration. By using the maximum principle and constructing an auxiliary function, we obtain the interior gradient estimate of $(k-1)$-admissible solutions.
Abstract : Let $\mathcal{A}$ be the collection of analytic functions $f$ defined in $\mathbb{D}:=\left\{\xi\in\mathbb{C}:|\xi|<1\right\}$ such that $f(0)=f'(0)-1=0$. Using the concept of subordination ($\prec$), we define \mathcal{S}^*_{\ell}:= \left\{f\in\mathcal{A}:\frac{\xi f'(\xi)}{f(\xi)}\prec\Phi_{\scriptscriptstyle{\ell}}(\xi)=1+\sqrt{2}\xi+\frac{\xi^2}{2},\;\xi\in\mathbb{D}\right\},where the function $\Phi_{\scriptscriptstyle{\ell}}(\xi)$ maps $\mathbb{D}$ univalently onto the region $\Omega_{\ell}$ bounded by the limacon curve \left(9u^2+9v^2-18u+5\right)^2-16\left(9u^2+9v^2-6u+1\right)=0.For $0
Abstract : In [4], the authors showed that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$ and $h_i\le h_1$ is a Gorenstein sequence, then $h_1=h_i$ for every $1\le i\le e-1$ and $e\ge 6$. In this paper, we show that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $h_2=4e-3$, and $h_i\le h_2$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 7$. We also propose an open question that if an $h$-vector $(h_0,h_1,\dots,h_e)$ with $h_1=4e-4$, $4e-3<h_2\le {(h_1)_{(1)}}| ^{+1}_{+1}$, and $h_2\le h_i$ is a Gorenstein sequence, then $h_2=h_i$ for every $2\le i\le e-2$ and $e\ge 6$.
Abstract : In this paper, the existence and uniqueness for the global solution of neutral stochastic functional differential equation is investigated under the locally Lipschitz condition and the contractive condition. The implicit iterative methodology and the Lyapunov-Razumikhin theorem are used. The stability analysis for such equations is also applied. One numerical example is provided to illustrate the effectiveness of the theoretical results obtained.
Abstract : For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first $k$ successive minima for three dimensional lattices for each $k\in\{1,2,3\}$ by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first $k$ shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For $\delta\ge 0.9$, we prove that LLL($\delta$) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.
Abstract : In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with generalized Killing structure Jacobi operator.
Abstract : Let $G$ be a non-discrete locally compact abelian group, and $\mu$ be a transformable and translation bounded Radon measure on $G$. In this paper, we construct a Segal algebra $S_{\mu}(G)$ in $L^1(G)$ such that the generalized Poisson summation formula for $\mu$ holds for all $f\in S_{\mu}(G)$, for all $x\in G$. For the definitions of transformable and translation bounded Radon measures and the generalized Poisson summation formula, we refer to L. Argabright and J. Gil de Lamadrid's monograph in 1974.
Abstract : For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.
J. Korean Math. Soc. 2022; 59(1): 193-204
https://doi.org/10.4134/JKMS.j210357
Jiaogen Zhang
J. Korean Math. Soc. 2022; 59(1): 13-33
https://doi.org/10.4134/JKMS.j200626
Jung Hee Cheon, Yongha Son, Donggeon Yhee
J. Korean Math. Soc. 2022; 59(1): 35-51
https://doi.org/10.4134/JKMS.j200650
J. Korean Math. Soc. 2022; 59(1): 151-170
https://doi.org/10.4134/JKMS.j210258
Jiaogen Zhang
J. Korean Math. Soc. 2022; 59(1): 13-33
https://doi.org/10.4134/JKMS.j200626
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Jundong Zhou
J. Korean Math. Soc. 2022; 59(1): 53-69
https://doi.org/10.4134/JKMS.j200665
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
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